Problem: $\dfrac{ 5x - 7y }{ 6 } = \dfrac{ 10x + 5z }{ -6 }$ Solve for $x$.
Notice that the left- and right- denominators are opposite $\dfrac{ 5x - 7y }{ {6} } = \dfrac{ 10x + 5z }{ -{6} }$ So we can multiply both sides by $6$ ${6} \cdot \dfrac{ 5x - 7y }{ {6} } = {6} \cdot \dfrac{ 10x + 5z }{ -{6} }$ $5x - 7y = - \cdot \left( 10x + 5z \right) $ Distribute the negative sign on the right side. $5x - 7y = -10x - 5z$ ${5}x - {7}y = -{10}x - {5}z$ Combine $x$ terms on the left. ${5x} - 7y = -{10x} - 5z$ ${15x} - 7y = -5z$ Move the $y$ term to the right. $15x - {7y} = -5z$ $15x = -5z + {7y}$ Isolate $x$ by dividing both sides by its coefficient. ${15}x = -5z + 7y$ $x = \dfrac{ -5z + 7y }{ {15} }$